Introduction to Coordinate Geometry

Class 9Coordinate Geometry

Coordinate Geometry is the branch of mathematics that uses algebraic equations to describe geometric shapes and positions on a plane.

The French mathematician René Descartes developed this system in the 17th century by combining algebra and geometry. In his honour, the coordinate plane is also called the Cartesian Plane.

In Class 9, coordinate geometry introduces the method of locating a point in a plane using two perpendicular number lines called axes. Every point on the plane is represented by an ordered pair of numbers.

This chapter forms the foundation for more advanced topics such as the distance formula, section formula, and equation of a line studied in Class 10 and beyond.

What is Introduction to Coordinate Geometry?

Definition: Coordinate Geometry is the study of geometry using a coordinate system, where each point in the plane is identified by an ordered pair of numbers (x, y).

The Cartesian Coordinate System consists of:

A horizontal number line called the x-axis
A vertical number line called the y-axis
The point of intersection of the two axes, called the origin, denoted by O(0, 0)

Key Terms:

Abscissa: The x-coordinate of a point (horizontal distance from the y-axis)
Ordinate: The y-coordinate of a point (vertical distance from the x-axis)
Ordered Pair: A pair (x, y) where x is the abscissa and y is the ordinate
Quadrant: One of the four regions into which the axes divide the plane

The Four Quadrants:

Quadrant I (Q-I): x > 0, y > 0 — both coordinates positive
Quadrant II (Q-II): x < 0, y > 0 — x negative, y positive
Quadrant III (Q-III): x < 0, y < 0 — both coordinates negative
Quadrant IV (Q-IV): x > 0, y < 0 — x positive, y negative

Points on the Axes:

A point on the x-axis has coordinates of the form (x, 0)
A point on the y-axis has coordinates of the form (0, y)
The origin has coordinates (0, 0)

Introduction to Coordinate Geometry Formula

Key Relationships:

1. Coordinates of the Origin:

O = (0, 0)

2. General form of a point in the Cartesian Plane:

P = (x, y)

Where:

x = abscissa (distance from y-axis, positive to the right, negative to the left)
y = ordinate (distance from x-axis, positive upward, negative downward)

3. Sign Convention by Quadrant:

Q-I: (+, +)
Q-II: (−, +)
Q-III: (−, −)
Q-IV: (+, −)

4. Mirror Reflections:

Reflection of (a, b) in the x-axis is (a, −b)
Reflection of (a, b) in the y-axis is (−a, b)
Reflection of (a, b) in the origin is (−a, −b)

Derivation and Proof

How the Cartesian Coordinate System Works:

Step 1: Draw two perpendicular number lines

The horizontal line is the x-axis (also called the abscissa axis).
The vertical line is the y-axis (also called the ordinate axis).
These two lines intersect at the origin O(0, 0).

Step 2: Mark equal units on both axes

On the x-axis, positive numbers go to the right and negative numbers to the left.
On the y-axis, positive numbers go upward and negative numbers downward.

Step 3: Locate a point P(a, b)

Start at the origin O.
Move a units along the x-axis (right if positive, left if negative).
From that position, move b units parallel to the y-axis (up if positive, down if negative).
The final position is the point P(a, b).

Step 4: Read coordinates of a plotted point

Draw a perpendicular from the point to the x-axis. The foot gives the x-coordinate.
Draw a perpendicular from the point to the y-axis. The foot gives the y-coordinate.
Write the coordinates as an ordered pair (x, y).

Step 5: Identify the quadrant

Check the signs of x and y to determine which quadrant the point lies in.
If either coordinate is zero, the point lies on an axis, not in any quadrant.

Types and Properties

Types of Points in the Cartesian Plane:

1. Points in Quadrants

These have both x and y coordinates non-zero.
The signs of x and y determine the quadrant.
Example: (3, 5) lies in Q-I; (−2, 4) lies in Q-II.

2. Points on the x-axis

Form: (x, 0) — the y-coordinate is always 0.
These points lie on the horizontal axis.
Example: (4, 0), (−3, 0).

3. Points on the y-axis

Form: (0, y) — the x-coordinate is always 0.
These points lie on the vertical axis.
Example: (0, 5), (0, −7).

4. The Origin

The unique point (0, 0) where both axes meet.
It does not belong to any quadrant.

5. Collinear Points

Three or more points are collinear if they lie on the same straight line.
Example: (1, 2), (2, 4), (3, 6) are collinear since they all satisfy y = 2x.

Solved Examples

Example 1: Example 1: Identify the quadrant

Problem: In which quadrant do the following points lie? (a) (3, 7) (b) (−4, 2) (c) (−5, −3) (d) (6, −1)

Solution:

(3, 7): x = 3 > 0, y = 7 > 0 ⇒ Quadrant I
(−4, 2): x = −4 < 0, y = 2 > 0 ⇒ Quadrant II
(−5, −3): x = −5 < 0, y = −3 < 0 ⇒ Quadrant III
(6, −1): x = 6 > 0, y = −1 < 0 ⇒ Quadrant IV

Answer: Q-I, Q-II, Q-III, Q-IV respectively.

Example 2: Example 2: Identify the axis

Problem: State whether the following points lie on the x-axis, y-axis, or neither. (a) (0, −8) (b) (5, 0) (c) (3, 4) (d) (0, 0)

Solution:

(0, −8): x = 0, so the point lies on the y-axis.
(5, 0): y = 0, so the point lies on the x-axis.
(3, 4): Both coordinates are non-zero, so it lies in Quadrant I (neither axis).
(0, 0): This is the origin (intersection of both axes).

Answer: y-axis, x-axis, neither (Q-I), origin.

Example 3: Example 3: Write coordinates from a description

Problem: A point P is 4 units to the left of the y-axis and 3 units above the x-axis. Write the coordinates of P.

Solution:

Given:

4 units to the left of the y-axis ⇒ x = −4
3 units above the x-axis ⇒ y = 3

Answer: The coordinates of P are (−4, 3). This point lies in Quadrant II.

Example 4: Example 4: Find the abscissa and ordinate

Problem: For the point Q(−7, 5), identify the abscissa and ordinate. State the quadrant.

Solution:

Abscissa (x-coordinate) = −7
Ordinate (y-coordinate) = 5
Since x < 0 and y > 0, the point lies in Quadrant II.

Answer: Abscissa = −7, Ordinate = 5, Quadrant = II.

Example 5: Example 5: Reflection in the x-axis

Problem: Find the reflection of the point A(3, −5) in the x-axis.

Solution:

Rule: Reflection of (a, b) in the x-axis is (a, −b).

A = (3, −5)
Reflection = (3, −(−5)) = (3, 5)

Answer: The reflection of A(3, −5) in the x-axis is (3, 5).

Example 6: Example 6: Reflection in the y-axis

Problem: Find the reflection of the point B(−2, 7) in the y-axis.

Solution:

Rule: Reflection of (a, b) in the y-axis is (−a, b).

B = (−2, 7)
Reflection = (−(−2), 7) = (2, 7)

Answer: The reflection of B(−2, 7) in the y-axis is (2, 7).

Example 7: Example 7: Identify coordinates from position

Problem: A point R is 6 units to the right of the origin and 2 units below the x-axis. Write its coordinates and quadrant.

Solution:

6 units to the right ⇒ x = 6
2 units below the x-axis ⇒ y = −2

Answer: R = (6, −2), lying in Quadrant IV.

Example 8: Example 8: Points on axes

Problem: Write the coordinates of three points that lie on the x-axis and three points that lie on the y-axis.

Solution:

Points on the x-axis (y = 0):

(2, 0), (−5, 0), (10, 0)

Points on the y-axis (x = 0):

(0, 3), (0, −4), (0, 8)

Answer: Any point on the x-axis has the form (x, 0); any point on the y-axis has the form (0, y).

Example 9: Example 9: Distance from the axes

Problem: Find the perpendicular distance of the point S(−3, 4) from (a) the x-axis and (b) the y-axis.

Solution:

The perpendicular distance of a point (x, y) from the x-axis = |y| = |4| = 4 units
The perpendicular distance of a point (x, y) from the y-axis = |x| = |−3| = 3 units

Answer: Distance from x-axis = 4 units; Distance from y-axis = 3 units.

Example 10: Example 10: Plot and identify a shape

Problem: Plot the points A(1, 1), B(4, 1), C(4, 5), D(1, 5) on the Cartesian plane and identify the shape ABCD.

Solution:

Step 1: Plot each point:

A(1, 1) — Q-I
B(4, 1) — Q-I
C(4, 5) — Q-I
D(1, 5) — Q-I

Step 2: Join A→B→C→D→A.

Step 3: Verify:

AB = 4 − 1 = 3 units (horizontal)
BC = 5 − 1 = 4 units (vertical)
CD = 4 − 1 = 3 units (horizontal)
DA = 5 − 1 = 4 units (vertical)
Opposite sides are equal, and all angles are 90°.

Answer: ABCD is a rectangle with length 4 units and breadth 3 units.

Real-World Applications

Applications of Coordinate Geometry:

Maps and Navigation: GPS systems use coordinates (latitude and longitude) to locate positions on Earth. Every location is described by an ordered pair of numbers.
Computer Graphics: Images on screens are created by assigning colour values to pixels identified by (x, y) coordinates. Animation relies on transforming coordinates frame by frame.
Architecture and Engineering: Blueprints and CAD software use coordinate systems to specify exact positions of structural elements.
Physics: Motion is analysed by plotting position-time and velocity-time graphs using the Cartesian plane.
Data Visualization: Bar graphs, line graphs, and scatter plots use the x-y plane to represent data visually.
Robotics: Robots navigate spaces using coordinate geometry to determine paths and avoid obstacles.

Key Points to Remember

The Cartesian Plane consists of two perpendicular number lines — the x-axis (horizontal) and the y-axis (vertical).
The point where the axes intersect is the origin O(0, 0).
Every point in the plane is represented by an ordered pair (x, y), where x is the abscissa and y is the ordinate.
The order matters: (3, 5) and (5, 3) are different points.
The axes divide the plane into four quadrants: Q-I (+, +), Q-II (−, +), Q-III (−, −), Q-IV (+, −).
Points on the x-axis have coordinates (x, 0); points on the y-axis have coordinates (0, y).
The perpendicular distance of (x, y) from the x-axis is |y| and from the y-axis is |x|.
Reflection of (a, b) in the x-axis is (a, −b); in the y-axis is (−a, b); in the origin is (−a, −b).
Coordinate geometry connects algebra and geometry, allowing geometric problems to be solved using equations.
René Descartes invented the coordinate system, which is why it is called the Cartesian system.

Practice Problems

Plot the following points on the Cartesian plane and identify the quadrant: A(2, 3), B(−4, 5), C(−1, −6), D(7, −2).
Write the coordinates of a point that lies on the negative x-axis, 8 units from the origin.
Find the reflection of the point (−6, −3) in (a) the x-axis, (b) the y-axis, (c) the origin.
A rectangle has vertices at P(2, 3), Q(7, 3), R(7, 8), and S(2, 8). Find the lengths of its sides.
Determine the perpendicular distance of the point (−9, 12) from both axes.
If the abscissa of a point is twice its ordinate, and the point lies in Q-I with ordinate 4, write the coordinates of the point.
Three vertices of a square are (1, 1), (5, 1), and (5, 5). Find the fourth vertex.
Plot the points (0, 3), (0, −3), (3, 0), (−3, 0). What shape do they form when joined?

Frequently Asked Questions

Q1. What is coordinate geometry?

Coordinate geometry is the branch of mathematics that uses a coordinate system (x-axis and y-axis) to describe the positions of points in a plane using ordered pairs (x, y). It connects algebra with geometry.

Q2. What is the Cartesian plane?

The Cartesian plane is a two-dimensional plane formed by two perpendicular number lines — the horizontal x-axis and the vertical y-axis — intersecting at the origin O(0, 0). It is named after René Descartes.

Q3. What are the four quadrants?

The x-axis and y-axis divide the plane into four quadrants: Q-I (+, +), Q-II (−, +), Q-III (−, −), Q-IV (+, −). The signs refer to the x and y coordinates of points in that region.

Q4. What is the difference between abscissa and ordinate?

The abscissa is the x-coordinate (horizontal distance from the y-axis). The ordinate is the y-coordinate (vertical distance from the x-axis). For point P(3, 7), the abscissa is 3 and the ordinate is 7.

Q5. Does the order of coordinates matter?

Yes. (3, 5) and (5, 3) are different points. The first number is always the x-coordinate and the second is the y-coordinate. This is why they are called ordered pairs.

Q6. Where does a point with coordinates (0, 5) lie?

A point (0, 5) lies on the y-axis, since its x-coordinate is 0. It is 5 units above the origin.

Q7. What are the coordinates of the origin?

The origin is the point where the x-axis and y-axis intersect. Its coordinates are (0, 0).

Q8. How do you find the distance of a point from the axes?

The perpendicular distance of (x, y) from the x-axis is |y| (the absolute value of the ordinate). The distance from the y-axis is |x| (the absolute value of the abscissa).

Q9. Who invented coordinate geometry?

René Descartes, a French mathematician and philosopher, invented the coordinate system in 1637. Pierre de Fermat independently developed similar ideas around the same time.

Q10. Is coordinate geometry part of the CBSE Class 9 syllabus?

Yes. Coordinate geometry is Chapter 3 in the CBSE Class 9 NCERT Mathematics textbook. It covers the Cartesian plane, ordered pairs, quadrants, and plotting of points.

Introduction to Coordinate Geometry

What is Introduction to Coordinate Geometry?

Introduction to Coordinate Geometry Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Example 1: Identify the quadrant

Example 2: Example 2: Identify the axis

Example 3: Example 3: Write coordinates from a description

Example 4: Example 4: Find the abscissa and ordinate

Example 5: Example 5: Reflection in the x-axis

Example 6: Example 6: Reflection in the y-axis

Example 7: Example 7: Identify coordinates from position

Example 8: Example 8: Points on axes

Example 9: Example 9: Distance from the axes

Example 10: Example 10: Plot and identify a shape

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is coordinate geometry?

Q2. What is the Cartesian plane?

Q3. What are the four quadrants?

Q4. What is the difference between abscissa and ordinate?

Q5. Does the order of coordinates matter?

Q6. Where does a point with coordinates (0, 5) lie?

Q7. What are the coordinates of the origin?

Q8. How do you find the distance of a point from the axes?

Q9. Who invented coordinate geometry?

Q10. Is coordinate geometry part of the CBSE Class 9 syllabus?

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